The usual formula for calculating the area of a triangle is taking half the product of its base and altitude. This, of course, requires first calculating the altitude. That's why I was happy when I discovered Heron's formula for calculating the area of an arbitrary triangle from the length of its sides. I wrote about Heron's formula in a previous article (Random Triangles, January 26, 2017).

Hero's simple formula for calculating the area A of an arbitrary triangle from the length of its sides appears in his Metrica (c. 60 AD).[1] Using a, b, and c as the length of the sides, the formula is as follows:

Langmuir's simple model of adsorption, which concerned just gas adsorption on a surface, supposed that gas molecules would stick to the surface by chemical adsorption or physical adsorption. He also supposed that the adsorbed film would be one molecule thick. He calculated that at a constant temperature (thus, the isotherm designation) the surface coverage θA would follow the simple expression, θA = P/(P + P0), in which P is the partial pressure of the gas, and P0 is a critical pressure. The graph illustrates how the surface coverage levels off at pressures greater than P0, in this example, 50.

"We often find inspiration in nature, and plants have discovered the best way to absorb chemicals such as carbon dioxide from their environment... In this case, we used that idea but at a much, much smaller scale - about one-millionth the size, in fact."[3]

The electrode, created by a two-step microwaveplasmachemical vapor deposition process, has highly oriented carbon nanotube array walls decorated by graphene.[2] The nanotubes allow a fast diffusion of ions during charge/discharge cycles, while the graphene petals significantly enhance mechanical robustness of the assemblage.[2-3] The hollow, cylindrical carbon nanotubes, are about 20 to 30 nanometers in diameter, and the graphene petal-like structures are about 100 nanometers wide.[3] The tunnel-shaped arrays allow ions to transport stored energy flow with much less resistance between the electrolyte and the surface than if the electrode surfaces were flat.[3]

The electrode design provides the same amount of energy storage as similar electrodes, but it's smaller and lighter. Experiments showed that it produced 30 percent better capacitance for its mass than similar carbon materials, and 30 times better capacitance per area. For these reasons it produced 10 times more power and retained 95 percent of its initial capacitance after more than 10,000 charging cycles (see graphs).[3] The areal capacitance was 2.35 Farad/cm2, which works out to be about 500 Farads per gram, and there was a capacitance retention of about 95% over 10,000 cycles.[2-3]